LECTURE 6 (202 b 30-203 b 14)

PHYSICS CONSIDERS THE INFINITE. THE OPINIONS OF THE ANCIENTS CONCERNING THE INFINITE

             326. After the Philosopher has treated motion, he begins here to treat the infinite.

             First he shows that it pertains to natural science to treat the infinite. Secondly, where he says, 'Belief in the existence . . .' (203 b 15), he begins to treat the infinite.

             Concerning the first part he makes two points. First he shows that it pertains to natural science to treat the infinite. Secondly, where he says, '. . . and indeed to a man . . .' (203 a 2), he sets forth the opinions of the ancient philosophers concerning the infinite.

             327. He explains the first point both with an argument and with an example.

             The argument is as follows. Natural science deals with magnitudes and time and motion. But in these things either the finite or the infinite must be found. For every magnitude or motion or time is contained under one of these, i.e., under the finite or the infinite. Therefore, it pertains to natural science to consider the infinite in terms of whether it is and what it is.

             But someone might say that the consideration of the infinite belongs to first philosophy because of its commonness. To overcome this objection he adds that it is not necessary for every being to be finite or infinite. For a point and a passion, i.e., a passive quality, are contained under neither of these. However, that which pertains to the consideration of first philosophy follows upon being insofar as it is being, and not upon some determinate genus of being.

             328. Next where he says, 'The appropriateness . . .' (202 b 37), he explains the same thing with an example taken from the considerations of the natural philosophers.

             For all who have reasonably treated philosophy of this sort, i.e., natural philosophy, have made some mention of the infinite. From this there arises a probable argument from the authority of wise men to the effect that natural philosophy should treat the infinite.

             329. Next where he says, '. . . and indeed to a man . . .' (203 a 2), he sets forth the opinions of the ancients concerning the infinite. First he shows how they differ. Secondly, where he says, 'Nor is it without reason . . .' (203 b 3), he shows how they agree.

             Concerning the first part he makes two points. First he sets forth the opinions about the infinite of those who were not philosophers of nature, i.e., the Pythagoreans and the Platonists. Secondly, where he says, 'The physicists, on the other hand . . .' (203 a 16), he sets forth the opinions of the natural philosophers.

             Concerning the first part he makes two points. First he shows in what respect the Pythagoreans and Platonists agree; and secondly, in what respect they differ, where he says, 'Only the Pythagoreans . . .' (203 a 6).

             330. He says, therefore, first that all the philosophers held that the infinite is some sort of a principle of beings. But it was peculiar to the Pythagoreans and the Platonists that they held that the infinite is not an accident of some other nature, but is a certain thing existing in itself. And this was consistent with their opinion because they held that numbers and quantities are the substances of things. But the infinite is in quantity. Hence they held that the infinite is something existing in itself.

             331. Next where he says, 'Only the Pythagoreans . . .' (203 a 6), he explains the differences between Plato and the Pythagoreans, first with reference to the position of the infinite, and secondly, with reference to its source, where he says, 'Further the Pythagoreans . . .' (203 a 10).

             With reference to the position of the infinite Plato differed from the Pythagoreans in two ways.

             The Pythagoreans posited an infinite only in sensible things. For the infinite belongs to quantity, and the first quantity is number. Hence the Pythagoreans did not posit number as separate from sensible things. Rather they said that number is the substance of sensible things. And as a result the infinite exists only in sensible things.

             Furthermore, Pythagoras thought that sensible things which are under the heavens are circumscribed by the heavens. Hence there can be no infinite in these things. And because of this he held that the infinite exists in sensible things outside of the heavens.

             But Plato, on the contrary, held that nothing is outside the heavens. He did not say that there is any sensible body outside of the heavens, because he said that the heavens contain all sensible things. And he did not even say that the ideas and species of things, which he held to be separated, are outside of the heavens. For 'inside' and 'outside' signify place. But according to him the ideas are not in any place, because place pertains to corporeal things.

             Furthermore, Plato said that the infinite is not only in sensible things but also in the separated ideas. For even in the separated numbers themselves there is something formal, as the one, and something material, as two, from which every number is composed.

             332. Next where he says, 'Further the Pythagoreans . . .' (203 a 10), he explains the difference between their opinions with reference to the source of the infinite. He says that the Pythagoreans referred the infinite to one source, namely, to even number. They explained this in two ways.

             First, with an argument. That which is encompassed by another and terminated by another, insofar as it is in itself, has the nature [ratio] of the infinite. But what encompasses and terminates has the nature [ratio] of a terminus. Now even number is comprehended and encompassed under the odd. For if some even number is proposed, it clearly is divisible. When, however, by the addition of unity it is reduced to an odd number, a certain indivision follows, as if the even were restricted by the odd. Hence it seems that the even is infinite in itself, and causes infinity in others.

             He explains the same thing by means of an example. As evidence of this we must note what happens in geometry. A gnomon is said to be a square erected on the diagonal by the addition of two lines. Thus a

IMAGE #1

gnomon of this sort surrounding a square forms a square. Therefore by comparing this to numbers gnomons can be said to be numbers which are added to other numbers.

             Moreover, it must be noted that if one takes odd numbers according to the order of natural progression, and if one adds the first odd number, i.e., three, to unity which is virtually square (insofar as one is a unit), then four is produced, which is a square number, for twice two is four. If, next, to this second square the second odd number, i.e., five, is added, then nine is formed which is the square of three, for three times three is nine. If, further, to this third square the third odd number, i.e., seven, is added, then sixteen is formed which is the square of four. And thus by an ordered addition of odd numbers, the same form of number always results, i.e., a square.

             However, a different figure always results from the addition of even numbers. For if the first even number, i.e., two, is added to one, then three is produced, which is a three sided figure. If, next, the second even number, i.e., four, is added, then seven is produced, which is a heptagonal figure. And thus the figure is always varied by the addition of even numbers.

             And this seems to be a sign that uniformity pertains to odd numbers, whereas dissimilarity and variety and the infinite pertain to even numbers.

             And this is what he says. That which occurs in numbers is an indication that the infinite is consequent upon even number. When gnomons surround the one, i.e., when numbers are added to one, and are outside, i.e., are around other numbers, then sometimes one species is formed, i.e., one numerical form results from the addition of an even number, and sometimes another species is formed, i.e., by the addition of an odd number. And thus it is clear why Pythagoras attributes infinity to even number.

             Plato, however, attributed the infinite to two sources, namely, the great and the small. For according to him these two are on the side of matter, to which the infinite belongs.

             333. Next where he says, 'The physicists on the other hand . . .' (203 a 16), he sets forth the opinions of the natural philosophers concerning the infinite.

             It must be noted that all the natural philosophers, i.e., those who treated the principles of things naturally, said that the infinite is not subsistent in itself, as was said above. Rather they held that the infinite is an accident of some nature presupposed for it.

             Therefore, those who posited only one material principle, whatever it might be, of the things which are called elements, either air or water or some intermediate, said that this principle is infinite.

             None of those who made the elements many but finite in number held that the elements are infinite in quantity. For this distinction of elements seems to be contradicted by the infinity of any one of them.

             But those who made the elements infinite in number say that some one infinite thing comes to be from all these infinite elements by contact.

             334. These latter philosophers were Anaxagoras and Democritus, who differed on two points.

             [They differed] first with reference to the quiddity of the infinite principles. For Anaxagoras said that these infinite principles are infinite, similar parts, such as flesh and bone and things of this sort. Democritus, on the other hand, held that such infinite principles are indivisible bodies which differ according to figure. And he said that these bodies are the seeds of all nature.

             The other difference is with respect to the relation that these principles have to each other.

             Anaxagoras said that each of these infinite parts is mixed with every other part, so that in every part of flesh there would be bone, and vice versa. And the same applies to other things. He held this because he saw that everything comes to be from everything. And since he believed that everything which comes to be from something is in that thing, he concluded that everything is in everything.

             And from this he seems to affirm that at one time all things were fused together, and nothing was distinct from anything else. Thus, this flesh and this bone are mixed together, which is demonstrated by the fact that they are generated from each other. And it is the same with any other thing. Therefore, at one time all things were together. This is to posit a beginning of the distinction [of things] not only for some one thing, but for all things together. He proved this as follows.

             That which comes to be from another was previously mixed with that thing, and comes to be by reason of the fact that it is separated from it. But all things come to be, although not all at once. Therefore, it is necessary to posit one principle of generation for all things, and not just for each thing. And this one principle he called intellect to which alone it belongs to separate and to gather together, because of the fact that it is unmixed.

             Now that which comes to be through intellect seems to have some sort of a beginning. For an intellect works by beginning from a determinate principle. If, therefore, separation occurs because of an intellect, then it is necessary to say that separation would have some sort of a beginning. Hence he concluded that at one time all things were together, and that the motion by which things are separated from each other began at some time, since there was no motion prior to this. Therefore, Anaxagoras held that one principle comes to be from the other.

             But Democritus says that one principle does not come to be from another. However, the nature of body, which is common to all indivisible bodies, but differing according to parts and figures, is the principle of all things in respect to magnitude, insofar as he held that all divisible magnitudes are composed of indivisibles.

             And thus Aristotle concludes that it pertains to the natural philosopher to consider the infinite.

             335. Next where he says, 'Nor is it without reason . . .' (203 b 3), he sets forth four things upon which the ancient philosophers agreed concerning the infinite.

             The first of these is that all of them held that the infinite is a principle.

             And this was 'reasonable', i.e., they had a probable argument. For if the infinite exists, it is not possible that it be in vain, i.e., that it should not have some determinate position in beings. Nor can it have any power other than that of a principle. For everything in the world is either a principle or is from principles. But the infinite cannot have a principle, because that which has a beginning has an end. Hence it follows that the infinite is a principle.

             But it must be noted that in this argument they use 'principle' and 'end' equivocally. For that which is from a principle has a beginning of origin. But a principle and an end of quantity or magnitude is what is repugnant to the infinite.

             The second thing which they attributed to the infinite is that it is ungenerated and incorruptible. And this follows from the fact that it is a principle.

             For everything which comes to be must have an end as well as have a beginning. Moreover, there is an end to every corruption. However, an end is repugnant to the infinite. Hence to be generable and corruptible is repugnant to the infinite. And thus it is clear that there is no principle of the infinite, but rather the infinite is the principle of other things. And in this argument 'principle' and 'end' are again used equivocally, as they were above.

             The third thing which they attributed to the infinite is that it would contain and govern all things. For this seems to belong to the first principle.

             Those who did not posit causes other than matter (which they said is infinite), i.e., agents, such as the intellect which Anaxagoras posited, and love which Empedocles posited, maintained this. For to contain and to govern pertain more to an agent principle than to matter.

             The fourth thing which they attributed to the infinite is that it is something divine. For they called everything divine which is immortal or incorruptible. Anaximander and many of the ancient natural philosophers held this.